The chirally rotated Schrodinger functionalat work

Mattia Dalla Brida, Tomasz Korzec,Mauro Papinutto, Pol Vilaseca

33rd International Symposium on Lattice Field Theory,14th of July 2015, Kobe, Japan.

Outline

Introduction:

The chirally rotated Schrodinger functional in N = 2 slides.

Topics:

Renormalization of the non-singlet local currents forNf = 2 + 1 non-perturbatively O(a) improved Wilson-fermions.

Towards the non-perturbative computation of the RG-runningof a complete basis of ∆F = 2 four-quark operators.

The chirally rotated Schrodinger functionalA chiral rotation to the Schrodinger functional (Sint ’05, ’10)

Given the isospin doublets ψ and ψ satisfying standard SF b.c.’s,we consider the chiral rotation,

ψ ≡ Rχ ≡ e iπ2 γ5

τ 3

2 χ, ψ ≡ χR ≡ χe iπ2 γ5

τ 3

2 .

The fields χ and χ satisfy the chirally rotated SF (χSF) b.c.’s,

Q+χ(x)|x0=0 = 0,

χ(x)Q+|x0=0 = 0,Q± ≡ 1

2 (1± iγ0γ5τ3),

which are invariant under the (rotated-)parity transformation,

P5 : χ(x)→ iγ0γ5τ3 χ(x), χ(x)→ −iχ(x) γ0γ5τ

3, x = (x0,−x).

Since R is a non-anomalous symmetry transformation of massless QCD,in the continuum we expect the universality relations,

〈O[ψ,ψ]〉SF = 〈O[Rχ, χR]〉χSF.

On the lattice with Wilson-fermions these relations hold among properlyrenormalized correlation functions, up to discretization effects!

The chirally rotated Schrodinger functionalRenormalization and O(a) improvement (Sint ’05, ’10)

For Wilson-fermions, the χSF b.c.’s are realized by fine-tuning afinite dim. 3 boundary counterterm (e.g. at x0 = 0)

χQ−χR−→ −iψγ5τ

3P−ψ,

⇒ breaks parity and flavour symmetry: its coefficient, zf (g0), canbe fixed by imposing parity/flavour symmetry restoration.

Automatic (bulk) O(a) improvement:

⇒ NO bulk O(a) effects for P5-even obs. (OP5−→ +O),

⇒ bulk O(a) effects are located in P5-odd obs. (OP5−→ −O).

Full O(a) improvement needs in practice the tuning of a couple ofO(a) boundary counterterms. PT seems to be good! (Sint, Vilaseca ’12, ’14)

The set-up has been recently studied to 1-loop order in perturbationtheory, and for Nf = 2 dynamical fermions: (Sint, Vilaseca ’14; Sint, MDB ’14)

X Automatic O(a) improvement and universality.

X Competitive determinations of several finite renormalizationconstants.

Renormalization in the χSFThe correlation functions we need . . . (Leder, Sint ’10)

SF:

fX (x0) = − 12 〈ψf1 (x)ΓXψf2 (x)Of2f1

5 〉,kY (x0) = − 1

6

∑k

〈ψf1 (x)ΓYkψf2 (x)Of2f1

k 〉,

χSF:

g f1f2X (x0) = − 1

2 〈χf1 (x)ΓXχf2 (x)Qf2f15 〉,

l f1f2Y (x0) = − 16

∑k

〈χf1 (x)ΓYkχf2 (x)Qf2f1

k 〉,

where,

X = A0,V0,P,S ,

Yk = Ak ,Vk ,T0k , T0k .

and,

f1, f2 = u, d , u′d ′, f1 6= f2.

x0 = T

x0 = 0

C ′

C

L× L× L

X, Y

T

Bilinears of boundaryquark-fields

Of1f25 ,Of1f2

kR−→ Qf1f2

5 ,Qf1f2k

Renormalization in the χSFRenormalization conditions from universality relations

Universality relations:

We consider P5-even correlation functions: (V ≡ conserved current)

(fA)R = (guu′

A )R = (−igudV )R ⇒ ZA guu′

A = −igudV

+ O(a2),

(kV )R = (luu′

V )R = (−iludA )R ⇒ ZA ludA = iluu

′

V+ O(a2).

Renormalization conditions: (Leder, Sint ’10; Sint, MDB ’14)

The universality relations suggest to us simple renormalization conditionsfor the definition of ZA, e.g.,

Z gA ≡

−igudV

(x0)

guu′A (x0)

∣∣∣∣x0=

T2

, Z lA ≡

iluu′

V(x0)

ludA (x0)

∣∣∣∣x0=

T2

.

N.B.: The ZA’s so obtained are fully O(a) improved:

NO need for O(a) operator improvement i.e. cA(g0) or cV (g0).

O(a) boundary effects cancel out in the ratios.

Renormalization in the χSFRenormalization conditions from universality relations

Universality relations:

We consider P5-even correlation functions: (V ≡ conserved current)

(fA)R = (guu′

A )R = (−igudV )R ⇒ ZV gud

V = gudV

+ O(a2),

(kV )R = (luu′

V )R = (−iludA )R ⇒ ZV luu′

V = luu′

V+ O(a2).

Renormalization conditions: (Leder, Sint ’10; Sint, MDB ’14)

The universality relations suggest to us simple renormalization conditionsfor the definition of ZV , e.g.,

Z gV ≡

gudV

(x0)

gudV (x0)

∣∣∣∣x0=

T2

, Z lV ≡

luu′

V(x0)

luu′

V (x0)

∣∣∣∣x0=

T2

.

N.B.: The ZV ’s so obtained are fully O(a) improved:

NO need for O(a) operator improvement i.e. cV (g0) or cV (g0).

O(a) boundary effects cancel out in the ratios.

Renormalization in the χSFRenormalization conditions from universality relations

Universality relations:We consider P5-even correlation functions:

(fP)R = (iguu′

S )R = (gudP )R ⇒ ZS ig

uu′

S = ZPgudP + O(a2),

(kT )R = (iluu′

T)R = (ludT )R ⇒ ZT iluu

′

T= ZT l

udT + O(a2).

Renormalization conditions: (Leder, Sint ’10; Sint, MDB ’14)

The universality relations suggest to us simple renormalization conditionsfor the definition of ZP/ZS , or ZT/ZT , e.g.,

ZP

ZS≡ iguu′

S (x0)

gudP (x0)

∣∣∣∣x0=

T2

,ZT

ZT

≡iluu

′

T(x0)

ludT (x0)

∣∣∣∣x0=

T2

.

N.B.: The finite ratios so obtained are fully O(a) improved:

NO need for O(a) operator improvement i.e. cT (g0) or cT (g0).

O(a) boundary effects cancel out in the ratios.

Adding strangeness . . .Lattice action and other details

Question: How do we get Nf = 2 + 1?

Answer: We consider 2 χSF + 1 SF Wilson-fermions. (Sint ’10)

What’s new?

O(a) improved determinations now require an improved bulk action.

⇒ This has to be considered in any case (s. below)!

Twice as many O(a) boundary counterterm coefficients to be tuned.

⇒ They all do not contribute to ZA,ZV , . . ., at O(a), we are grand!

If renormalization conditions are defined only in terms of χSF fields,NO need for O(a) operator improvement!

Lattice action (cf. CLS) (S. Schaefer’s talk; Bruno, et. al. ’14)

Fermionic action: Nf=2+1 NPT O(a) improved Wilson-fermions.

Gauge action: Tree-level O(a2) improved Luscher-Weisz action.

χSF-specific: T = L; C = C ′ = 0; ct @ 1-loop; ds , ct @ tree-level.

Line of constant physicsRenormalization conditions in a fixed topological sector

LCP:

β Ltrg/a L/a

3.40 8 6, 8, 10, 123.46 9.04 6, 8, 10, 123.55 10.76 8, 10, 12, 163.70 13.89 8, 10, 12, 16

β tsym0 /a2 a (fm)

3.40 2.8468(61) ≈ 0.083.46 3.635(31) ≈ 0.073.55 5.150(23) ≈ 0.063.70 8.555(23) ≈ 0.05

Renormalization conditions: (Fritzsch, Ramos, Stollenwerk ’14)

mcrit : mPCAC =∂0g

udA,0(x0)

2gudP,0(x0)

∣∣∣∣x0=

T2

!= 0; zf : gud

A,0(x0)|x0=

T2

!= 0;

Z gA ≡

−igudV ,0

(x0)

guu′A,0 (x0)

∣∣∣∣x0=

T2

, 〈O0〉 ≡ 〈O · δQc ,0〉.

NOTE: Qc is the topological charge defined through the gradient flow,evaluated at c =

√8t/L = 0.6.

Determination of ZA at β = 3.55

0.772

0.78

0.0025 0.0065 0.0105 0.0145

(a/L)2

ZgA

ZlA

Determination of ZV at β = 3.70

0.758

0.764

0.0025 0.0065 0.0105 0.0145

(a/L)2

ZgV

ZlV

Renormalization constantsComparison between two different definitions of ZA as a function of g 2

0

0.76

0.78

1.62 1.66 1.7 1.74 1.78

ZA

g20

ZgA

ZlA

Renormalization constantsComparison between two different definitions of ZV as a function of g 2

0

0.72

0.74

0.76

1.62 1.66 1.7 1.74 1.78

ZV

g20

ZgV

ZlV

Test of universalityContinuum limit extrapolations of ZV ,A differences

0

0.005

0.01

0.015

0 0.004 0.008 0.012 0.016

(a/L)2

ZgV - Z

lV

ZgA - Z

lA

Results for ZP/ZS

0.8

0.83

0.86

0.0025 0.0065 0.0105 0.0145

ZP /

ZS

(a/L)2

β=3.55

β=3.70

Renormalization of ∆F = 2 four quark-operatorsFirst steps towards the non-perturbative running in the Nf = 2 + 1 theory?

Start: The χSF allows for finite-volume, mass-independentrenormalization schemes, compatible with auto. O(a) improvement.

⇒ Step 1. Identify valuable schemes within PT; both in terms ofanomalous dimensions, as well as cutoff effects in the SSFs. (P. Vilaseca’s talk)

⇒ Step 2. Feasibility study for a NPT determination of the runningusing Nf = 2 NPT O(a) improved Wilson-fermions. (Della Morte et. al. ’05)

Renormalization conditions:

Gi ;α = N−1α

⟨O′5QiO5

⟩Li ;α = N−1

α

⟨O′k QiOk

⟩Nα = gα1

1 lα21 gα2

Vlα3

V

(Z22 Z23

Z32 Z33

)(G2;α L2;β

G3;α L3;β

)=

(G2;α L2;β

G3;α L3;β

)g0=0

.

Step-scaling function (SSF):

σ(u) = lima→0

Σ(u, a/L), Σ(u, a/L) = Z(g0, 2L/a)Z−1(g0, L/a)∣∣u=g2(L)

.

Renormalization of Q+1

Continuum limit extrapolation for the lattice step-scaling function for g 2(L) = 3.3

1.155

1.175

1.195

1.215

0 0.01 0.02 0.03

Σ1(3

.3,a

/L)

(a/L)2

Linear

Constant

Renormalization of Q−2 , and Q−3Continuum limit extrapolations for the lattice step-scaling functions for g 2(L) = 3.3

1.198

1.218

1.238

0 0.01 0.02 0.03

Σ22(3

.3,a

/L)

(a/L)2

Linear

Constant-0.435

-0.42

-0.405

0 0.01 0.02 0.03

Σ23(3

.3,a

/L)

(a/L)2

Linear

Constant

0.17

0.175

0.18

0 0.01 0.02 0.03

Σ32(3

.3,a

/L)

(a/L)2

Linear

Constant

0.55

0.56

0.57

0 0.01 0.02 0.03

Σ33(3

.3,a

/L)

(a/L)2

Linear

Outlook

Outlook:

Little extra work is needed to finalize the results for ZV , and ZA.

The ensembles generated can be used for several other purposes:ZP/ZS , low-energy matchings, . . .

Still a lot to do and to understand for the non-perturbative runningof four-quark operators:

⇒ crucial to understand whether perturbation theory at NLOprovides a good picture of the high-energy regime we can reach.

⇒ make use of several different schemes to improve thedetermination of the RGI-operators.

. . . while we are thinking computers are running!

Investigate other applications of the χSF, as for example thedetermination of some improvement coefficients.